An integer is selected at random from the set of integers {1, 2, 3, ..

Question

An integer is selected at random from the set of integers {1, 2, 3, ..., 1000}. What is the probability that it is divisible by 4 but neither by 5 nor by 7?

A. 0.16
B. 0.162
C. 0.172
D. 0.175
E. 0.18

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geetsuri123 · Accepted Answer

when we write the series of multiples of 4, i.e - 4, 8, 12, 16, 20, 24,28, 32, 36, 40... we can see that in the first 10 multiples of 4, there are only 3 numbers that are divisible by 5 or 7, rest all are only divisible by 4. Moreover, we can extrapolate this finding and say that -

for every 10 consecutive multiples of 4, there are 3 numbers which are divisible by 5 or 7

Now, Total number of multiples of 4 in the A.P. series 4, 8, 12 .... till 1000 can be found out by -

T = a + (n-1)d 
1000 = 4 + (n-1)4 
n = 250

Now applying the unitary method, 
In 10 consecutive terms, there are 3 numbers divisible by 5 and 7. 
Hence in 250 consecutive terms, there are (250*3)/10 = 75 numbers divisible by 5 and 7.

Hence, number of terms which are divisible by 4 but are not divisible by 5 and 7 = 250 - 75 = 175

Total number of terms in the series 1, 2, 3...1000 = 1000

Hence, probability of selecting a number which is divisible by 4 but is not divisible by 5 and 7 = 175/1000 = 0.175

Answer - Option (C)

Iq · Answer

No. of multiples of 4 : 250

No. of multiples of 4 and 5 i.e. multiples of 20: 50

No. of multiples of 4 and 7 i.e. multiples of 28: 35

No. of multiples of 4, 5 and 7 i.e. multiples of 140: 7

So, multiples of 4 but not multiples of 5, 7: 250 - (50 + 35 - 7) = 172

Probability of multiples of 4 but not multiples of 5, 7 from 1 - 1000 = 172 /1000 =  0.172

Ans C

imeanup · Answer

A is divisible by 5.  P(A) =\frac{200}{1000}

B is divisible by 7.  P(B)=\frac{142}{1000}

For simplification, we are removing the   \frac{1}{1000}

P(A∪ B)=P(A)+P(B)-P(A∩B)=200+142-28=314

No of integers Not Divisible by  5 ,or, 7 =1000-314=686

Therefore the number of integers divisible by  4 =\frac{686}{4}=171.5  or,  0.172

Ans C

Note: Since a number is divisible by  5  and  7  if and only if it is divisible by  35  ( 5  and  7  are prime numbers),  P(AB) = \frac{28}{1000}

chiragarora001 · Answer

Is the answer C or D?
Since your result is matching the option D.

clinbb · Answer

P(4)=1000/4= 250 
P(4&5)= 1000/5= 50
P(4&7)= 1000/(4*7) = 35
P(4&7&5)=1000/(4*5*7) = 7

P(4) - P(4&5) - P(4&7)+ P(4&7&5)= 271 
271/1000

AJ0784 · Answer

No. of integers divisible by 4 = 1000/4=250
No. of integers divisible by 4&5 = 1000/20=50
No. of integers divisible by 4&7 = 1000/28=35
So no. of integers divisible by 4 but neither by 5 nor by 7 = 250-50-35=165
But it also includes integers divisible by both 20 & 28, therefore no. of such integers= 1000/140=7
We have to add 7 since 7 integers were subtracted twice from 250.
Therefore the total no. of integers divisible by 4 but neither by 5 nor by 7 = 165+7=172

Probability = 172/1000= 0.172

Rohanx9 · Answer

Bunuel ,

I'm having trouble discerning the wording on the "nor" component of the wording here:
"...divisible by 4 but neither by 5 nor by 7?"

To me, this sounds like it asking for:
"...divisible by neither 4, 5 AND 7"
i.e. shouldn't be divisible by 4*5*7 = 140. 
If its divisible by 4 and 5 = 20 its fine, or by 4 and 7 = 14 it's also fine. But it shouldn't be divisible by all three (120).

I think a clearer way to ask the question should be:
"...divisible by 4 but not 5, or not 7?"

Kinshook · Answer

An integer is selected at random from the set of integers {1, 2, 3, ..., 1000}.

What is the probability that it is divisible by 4 but neither by 5 nor by 7?

The number of integers divisible by 4 = 1000/4 = 250
The number of integers divisible by 4*5(20) = 1000/20 = 50
The number of integers divisible by 4*7(28) = 1000/28 = 35
The number of integers divisible by 4*7*5 (140) = 1000/140 = 7

The number of integers divisible by 4 and either 5 or 7 = 50 + 35 - 7 = 78
The number of integers divisible by 4 but neither by 5 nor by 7 = 250 - (78) = 172

The probability of selecting an integer is selected at random from the set of integers {1, 2, 3, ..., 1000} that is divisible by 4 but neither by 5 nor by 7 = 172/1000 = .172

IMO C

An integer is selected at random from the set of integers {1, 2, 3, ..

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An integer is selected at random from the set of integers {1, 2, 3, ..

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